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Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature \(\kappa \). For any point \(p\in M\) and any number \(\lambda \in {\mathbb {R}}\) with \(\lambda ^2+\kappa >0\), we show existence of a \(C^2\) spherical surface \(\mathcal {S}_\lambda (p)\) immersed in M with constant mean curvature \(\lambda \). Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group (Pansu, Conference on differential geometry on homogeneous spaces (Turin, 1983), pp 159–174, 1984) and the sub-Riemannian 3-sphere (Hurtado and Rosales, Math Ann 340(3):675–708, 2008). Then, we study variational properties of \(\mathcal {S}_\lambda (p)\) related to the area functional. First, we obtain uniqueness results for the spheres \(\mathcal {S}_\lambda (p)\) as critical points of the area under a volume constraint, thus providing sub-Riemannian counterparts to the theorems of Hopf and Alexandrov for CMC surfaces in Riemannian 3-space forms. Second, we derive a second variation formula for admissible deformations possibly moving the singular set, and prove that \(\mathcal {S}_\lambda (p)\) is a second order minimum of the area for those preserving volume. We finally give some applications of our results to the isoperimetric problem in sub-Riemannian 3-space forms.

Existence, characterization and stability of Pansu spheres in sub-Riemannian 3-space forms